*Simon Donaldson*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0007
- Subject:
- Mathematics, Geometry / Topology, Analysis

The genus of a compact oriented smooth surface S can be defined as one-half the dimension of the de Rham cohomology group H 1(S). Another way of defining the genus involves the ...
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The genus of a compact oriented smooth surface S can be defined as one-half the dimension of the de Rham cohomology group H 1(S). Another way of defining the genus involves the use of the Euler characteristic. This can be done via triangulations of the surface. This chapter describes this approach and then develops some applications.Less

The genus of a compact oriented smooth surface *S* can be defined as one-half the dimension of the de Rham cohomology group *H* ^{1}(*S*). Another way of defining the genus involves the use of the Euler characteristic. This can be done via triangulations of the surface. This chapter describes this approach and then develops some applications.

*Dusa McDuff and Dietmar Salamon*

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0002
- Subject:
- Mathematics, Analysis, Geometry / Topology

The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the ...
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The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.Less

The first chapter develops the basic concepts of symplectic topology from the vantage point of classical mechanics. It starts with an introduction to the Euler–Lagrange equation and shows how the Legendre transformation leads to Hamilton’s equations, symplectic forms, symplectomorphisms, and the symplectic action. It ends with a brief overview of some modern results in the subject on the symplectic topology of Euclidean space, such as the Weinstein conjecture and the Gromov nonsqueezing theorem.